MOTION OF A PARTICLE 125 9. Show that a rifle will shoot three times as high when its angle of elevation is 60° as when it is 30°, but will carry the same distance along a horizontal plane. 10. An emery wheel bursts into small pieces while making 100 revolutions per second. If the radius of the wheel is 10 cm. find farthest distance reached by any of its pieces. MOTION OF A PARTICLE UNDER THE ACTION OF A VARYING FORCE. 110. I. Uniform Circular Motion. — Consider the motion of a particle projected into a circular tube, Fig. 68, the inner surface of which is perfectly smooth. Let m be the mass of the particle, v its speed of projection and r the radius of the circle formed by the tube. The radius of the cross-section of the tube is supposed to be negligible. Suppose the particle to be acted upon by no forces except the reaction of the inner surface of the tube. Then, since the surface is smooth, the reaction is normal and consequently there is no force along the tangent to the path. Therefore the force equations for the tangential and the normal directions give dv ,,N FIG. 68. and m- =-N, r (2). where N represents the reaction of the walls of the tube upon the particle. It is evident from equation (1) that the magnitude of the velocity is constant. Therefore the particle describes the circular path with a constant speed. Equation (2) states that the normal reaction of the surface equals the.